dF/dT=4(aFc+bFl+eFr), F=heat flux, T=Temperature, c=conduction, l=latent, r=radiative and a,b and e are constants of proportionality.
That is one amazingly powerful equation. All three heat fluxes have different properties but adhere to the same laws of physics. So all heat fluxes share the same basic relationship with temperature. Only their rates of flow vary with the method and media of propagation. They share the inverse square law. A general relativity of flux.
The first time I say this formula, which I have since modified with the coefficients, was on the Blackboard blog. It is the result of a paper in an obscure and controversial journal, Energy and Environment
If the journal had not been E&E and the debate on the internet, that formula would greatly simplify solutions to common mixed heat flux problems in the atmosphere.
The debate seems to center around the "implicit" Planck response of the atmosphere, but really should be centered around the coefficients, a, b and especially e, which I will call the Effective emissivity of radiative flux at the surface of the Earth.
That emissivity is generally e={dF/dT}/4 if a one degree change in temperature is due to a 3.3Wm-2 change in flux, the e=3.3/4=0.825 a unit less value, as 3.3 is the actual flow to temperature and 4 is the ideal flow to temperature. Extremely simple coefficient. For conduction, a, would be the ratio of actual flow to ideal flow, b, actual flow to ideal flow. Incredibly simple and elegant mathematics.
The e value is implicit it the climate science literature, it ranges from 0.61 to 1, ideal, with often 0.926, the value estimated by Stefan-Boltzman for a real black body since an ideal black body does not exist.
Update: The value for e is still an issue. Why? I have no clue, it is a benchmark value that can be determined in many ways, if you compare the approximate greenhouse temperature 33 to the approximate greenhouse energy flux 155, 155/33=4.7 Wm-2/K 4/4.7= 0.851 This is the initial value I determined for e which allows for the effective emissivity of the atmosphere. The surface though is still not a true black body, water has a very high emissivity of 0.99, but the average emissivity of all of Earth's surface is 0.967, which multiplied by the 0.851=.825. If anyone has a better estimate let me know.
The elegance can be seen in the Earth's atmosphere. Heat flux at the surface is divided into three forms, each with different characteristics of flow. Radiant heat for the surface does not uniformly interact with the atmosphere, only a portion e, interacts at differing spectral lines and intensities.
So a 100W/m-2 flux upward, with e=0.825 would produce a radiative interaction impact 82.5 W/m-2 in the atmosphere, down welling longwave radiation for some. An effective temperature for many others. This is the Greenhouse Effect, due to radiative flux. Conduction, convection and latent heat fluxes all contribute with differing impact as defined by their coefficients. This is so simple it should not require a proof. It is obvious in our physical world.
Using data from the NASA energy budget drawing, the atmosphere absorbs, 24 conductive, 78 latent and 51 radiative Wm-2 from the surface for a total of 153, the greenhouse effect. The atmosphere also absorbs 68 W/m-2 from the sun, the combined atmospheric effect is 153+68=221. All of these are absorbed values, eFr, not Fr needing adjustment. Since the solar is absorbed at different altitudes, its effect propagates according to the same simple inverse squared relationship. How hard do you really what to make nature?
http://ourhydrogeneconomy.blogspot.com/2011/10/call-for-mathematicians-greenhouse.html
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